Are fruit on trees in public parks obligated in terumot and ma'aserot?

Answer

First, the trees are certainly NOT hefker, rather the fruit may be hefker. Practically speaking, this means that no one has the right to cut branches or harm the trees or uproot trees and transfer them elsewhere without asking permission from the municipality.

What about the fruit? What could possibly cause the fruit to be or not be hefker?

The municipality plants and cares for fruit trees (using municipal taxes) to add beauty to the area for the benefit of the local residents. It is possible that the municipality does not care if any individual – and that means anyone (including Arabs from other areas) – comes and pick fruit on the small or large scale; indeed, it actually could be doing the municipality a favor by eliminating the need to clean up all the fallen fruit. 

 It is possible, however, that they don't mind if locals pick, but don't want outsiders to pick the fruit. In this case, the fruit isn't actually hefker, since something isn't actually ownerless if you let one person but not another. 

Another possibility is that they are afraid that if people pick from the trees, they might harm them, and they actually don't want people to harvest the fruit.

 Best practice is if you want to pick (small or large scale) from trees in public parks, contact the head of the municipality's שפ"ע (urban renewal) department to ensure that it is ok. I asked the Givat Shmuel municipality about the fruit and was told that it is hefker, but I'm not sure if I asked the Beit Shemesh municipality. 

Note, though, that there are localities - yishuvim, kibbutzim, moshavim, where local fruit in public areas is meant for the locals and people actually mind if non-locals pick. In this case, the produce would be subject to terumot and ma'aserot.

 The same holds true for fruit trees growing in the shared yard of an apartment building or complex, where the local residents pay for their upkeep. In this case, the fruit is also not hefker and if you live there or have permission, you need to separate terumot and ma'aserot.

 Bottom line: Until you contact the municipality to find out if the fruit is actually hefker, when you pick fruit separate terumot and ma'aserot without a berachah (I'm assuming that you folks all live in Beit Shemesh).

And it's best to pick up the phone and contact the SHEFA department and ask if they mind you harvesting olives large-scale from the trees, and while you're at it, ask if they mind if non-local residents come and harvest them too. Then you will know if they are actually hefker and if they require the separation of terumot and ma'aserot. 

Again, if they let you since you are local, but don't allow others, then you would separate terumot and ma'aserot with a berachah.